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Pos-option

Pos-option-fix

Fixing function for pos-option structures.

Signature
(pos-option-fix x) → new-x
Arguments: x — Guard (pos-optionp x).
Returns: new-x — Type (pos-optionp new-x).

Definitions and Theorems

Function: pos-option-fix$inline

(defun pos-option-fix$inline (x)
  (declare (xargs :guard (pos-optionp x)))
  (let ((__function__ 'pos-option-fix))
    (declare (ignorable __function__))
    (mbe :logic (cond ((not x) nil)
                      (t (b* ((fty::val (pos-fix x))) fty::val)))
         :exec x)))

Theorem: pos-optionp-of-pos-option-fix

(defthm pos-optionp-of-pos-option-fix
  (b* ((new-x (pos-option-fix$inline x)))
    (pos-optionp new-x))
  :rule-classes :rewrite)

Theorem: pos-option-fix-when-pos-optionp

(defthm pos-option-fix-when-pos-optionp
  (implies (pos-optionp x)
           (equal (pos-option-fix x) x)))

Function: pos-option-equiv$inline

(defun pos-option-equiv$inline (x y)
  (declare (xargs :guard (and (pos-optionp x) (pos-optionp y))))
  (equal (pos-option-fix x)
         (pos-option-fix y)))

Theorem: pos-option-equiv-is-an-equivalence

(defthm pos-option-equiv-is-an-equivalence
  (and (booleanp (pos-option-equiv x y))
       (pos-option-equiv x x)
       (implies (pos-option-equiv x y)
                (pos-option-equiv y x))
       (implies (and (pos-option-equiv x y)
                     (pos-option-equiv y z))
                (pos-option-equiv x z)))
  :rule-classes (:equivalence))

Theorem: pos-option-equiv-implies-equal-pos-option-fix-1

(defthm pos-option-equiv-implies-equal-pos-option-fix-1
  (implies (pos-option-equiv x x-equiv)
           (equal (pos-option-fix x)
                  (pos-option-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: pos-option-fix-under-pos-option-equiv

(defthm pos-option-fix-under-pos-option-equiv
  (pos-option-equiv (pos-option-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-pos-option-fix-1-forward-to-pos-option-equiv

(defthm equal-of-pos-option-fix-1-forward-to-pos-option-equiv
  (implies (equal (pos-option-fix x) y)
           (pos-option-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-pos-option-fix-2-forward-to-pos-option-equiv

(defthm equal-of-pos-option-fix-2-forward-to-pos-option-equiv
  (implies (equal x (pos-option-fix y))
           (pos-option-equiv x y))
  :rule-classes :forward-chaining)

Theorem: pos-option-equiv-of-pos-option-fix-1-forward

(defthm pos-option-equiv-of-pos-option-fix-1-forward
  (implies (pos-option-equiv (pos-option-fix x) y)
           (pos-option-equiv x y))
  :rule-classes :forward-chaining)

Theorem: pos-option-equiv-of-pos-option-fix-2-forward

(defthm pos-option-equiv-of-pos-option-fix-2-forward
  (implies (pos-option-equiv x (pos-option-fix y))
           (pos-option-equiv x y))
  :rule-classes :forward-chaining)